#130029

order	512
nilpotency class	4
isoclinism	isoclinic class of #129789
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#60718	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

604 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,12,20,13,19,29,37,14,21,30,18,15,22,31,36)(11,23,26,39,32,17,28,34,27,38,24,33,25,35,40,16);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,28)(13,25)(14,27)(15,32)(17,20)(18,35)(19,33)(21,34)(22,39)(23,37)(24,31)(26,29)(30,40)(36,38);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,24)(13,32)(14,27)(15,25)(16,20)(18,34)(19,38)(21,35)(22,23)(26,30)(28,31)(29,40)(33,36)(37,39)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37225);